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Total No. <'lQuestions: 09]

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[Total No. of-Pages: 03

B. Tech. (Sem. - 1st/2nd)

ENGINEERING MATHEMATICS -I

SUBJECT CODE: AM -101 (2K4)

Paper ID : [AOlll](Note: Please fiIl subject code and paper'ID on OMR)

Time: 03 Hours

Instruction to Candidates:

1) Section -A is Compulsory.2) Attempt any Five questions from Section -B & C.

3) Selectat least TwoQuestionsfrom Section-B &: C,.

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Maximum Marks: 60

Section -A

a)

(2 Marks Each)

Find the equation of nonnal to the surface : r + y + z2 = a2.

b) Examine the convergence of 1...,.1.+.!..-1. + ~....234

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c) Define a homogeneous function.

d)r(m)r(n) . J 1 1

)If p(m, n)= rem + n) , then what IS /'"'l2' 2 . ?

e) M.l. of rectangular lamina about its side is =?

f)X2 y2 2z

Name the curve represented by : 2""- b2 =-a c

g)

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If (3 + X)3- (3 - X)3= 0, then prove that x = 3i tan ""3 r = 0, 1,2,.

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h) State DeMoivre's theorem.

i) What is ji = ?

j) State Ratio test.

Section - B

(8 Mar./t.sEach)

Q2) (a) Use method of Lagrange's to find the minimum value of;X} + yY + Z2,given that xyz = a3.

(b) Expand ex log(l + y) up to six terms of the Taylor series in theneighborhood of (0,0).

f . a(x,y,z) 2Q3) (a) I u = x + y+ z, uv = y + z, uvw = z show that ~I )

=u v.U\U,v,w

(b)

., X3+ 3 au auif u =tan-I y, prove that x-a + y A..= 2 sin 2x.

x + y tX vy

Q4) (a). Trace the curve ;x}/3+ y/3 = a2l3.

(b) Find the curvatureandradiusof curvatureof the curve:x = e - sin e, y = 1- cose .

Q5) (a) Show that the length of an arc of the cycloid:

x =a(9 - sin (}} y =a(l- cos (}) is 8a.

. 2 2

(b) Find the volume generated by revolving the ellipse: :6 + ~ =1aboutthex-axis.

Section - C(8 Marks Each)

Q6) (a) Find the equation of the cone whose vertex is (1,2,3) and which passesthrough the circle;x}+ y + z2 = 4, x + y + z =1.

(b) Find the centre and radius R of the circle ;x}+ Y + z2 -2y - 4z = 11,x+2y+2z=15.

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I Q7) (a) Change the order of integration in! = f f dydx and hence evaluate it.0 x2

4a

(b) Find the volume of the tetrahedron bounded by the coordinate axesand the plane x + y + z = a by triple integration.

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Q8) (a) Sumthe series: sin a + sin(a + /3)+ sin(a + 2/3)+ + sin(a + n -1/3).{/

. ~ I~b) If x + iy = eosh( u + iv) show that 2 + 2 =1.eosh v cosh u

2 3 X4. X x

Q9) (a) Find the intervalof eonvergenee of the series x - .J2+ .J3 ~ .J4 + 00.

(b) Test the convergence of the series:

(i) .JX3 +l-x.

123(n

oo) -+-+-+ 00

1.3 3.5' 5.7 ... .

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